# Tagged Questions

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### How to find $\int_0^\pi (\log(1 - 2a \cos(x) + a^2))^2 \mathrm{d}x, \quad a>1$?

Integration by parts is of no success. What else to try? $$\int_0^\pi (\log(1 - 2a \cos(x) + a^2))^2 \mathrm{d}x, \quad a>1$$
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### An integral involving the error function

I have in my notes the following problem. I recall it being quite difficult and needing a change of variables into polar or spherical coordinates. Assuming I have not made a typo, there is a nice ...
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### Trying to recall an integration trick

In my notes, I have the following problem. Find the volume of (a) $x^2+y^2 \le 1$, $x^2+z^2\le 1$ in $\mathbb R^3$ (b) $x^2+y^2 \le 1$, $x^2+z^2\le 1$, $y^2+z^2\le 1$ in $\mathbb R^3$ ...
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$$\int\cos x\cdot\cos^2(2x)\cdot\cos^3(3x)\cdot\cos^4(4x)\cdot\ldots\cdot\cos^{2002}(2002x)dx$$ Taken from the 2002 Romanian olympiad
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### Integrals from MIT integration bee

$\int\frac{dx}{2+2\sin x+\cos x}$ $\int_0^{\infty}\frac{\ln x}{1+x^2}dx$ $\int\frac{dx}{x(1+x^3)}$ In general what is $\int \frac{dx}{a+b\sin x}$?
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### how prove this integral inequality?

How prove that for all continuous and decreasing function $f:[0 ,1]\mapsto(0,+\infty)$ $$\frac{\int_{0}^1x(f(x))^2dx}{\int_{0}^1xf(x)dx}\leq \frac{\int_{0}^1(f(x))^2dx}{\int_{0}^1f(x)dx}$$ thanks in ...